I'd like to use a 12V battery to supply power to a 12V water pump (via a relay) and also power an Arduino connected to the same battery terminals in parallel. Would the drain from the battery when the pump is switched on (about 3A) damage the Arduino? I'm using a 10A Songle relay that can be switched with 5V from the Arduino.

sounds like a water pump with a brushed dc -
motor, so Type of battery ? ( to figure out the power output capability and its A.h capacity)

if it's a sealed lead acid-type then it can easily be done BUT you will need some filtering caps near the arduino and maybe another fat cap near the motor with a flyback diode. if your 3A @12V current draw exceeds what the battery can provide (battery dependant OR if the battery is flat) then it could switch off the arduino. if you have filtering and a flyback diode then you wont really damage the arduino, only risk is unwanted reset and switch-off.